\subsection{Submodularity of the problem}
Let $\mathcal{U} = \{(v_u,c_u);\, u\in\neigh{X}\}$ with $v_u=p_uw_u$ and
$c_u=p_u$.  Let $b\in\reals^+$ and $A\subseteq\mathcal{U}$, we denote by
$\mathcal{O}(A,b)$ the optimal solution to:
\begin{displaymath}
    \begin{split}
        &\max_{q\in[0,1]^n}\sum_{u\in\neigh{X}}q_uv_u\\
        &\text{s.t. }\sum_{u\in\neigh{X}}q_uc_u\leq b\text{ and } q_u=0\text{ if
    }u\notin A
\end{split}
\end{displaymath}
so that the optimization problem \eqref{eq:relaxed-multi} for relaxed
non-adaptive policies can be written:
\begin{equation}\label{eq:sub}
    \begin{split}
        &\max_{S\subseteq X}\mathcal{O}\big(\neigh{S},k-|S|\big)\\
        &\text{s.t. }|S|\leq k
    \end{split}
\end{equation}

In this section, we will show how \eqref{eq:sub} can be reduced to a submodular
maximization problem, leading to a fast combinatorial algorithm for the
adaptive seeding model in the voter model.

\begin{lemma}\label{lemma:nd}
    Let $A\subseteq\mathcal{U}$ and $x=(v,c)\in\mathcal{U}\setminus A$, then
    the function $b\mapsto \mathcal{O}(A\cup\{x\},b)-\mathcal{O}(A,b)$ is
    non-decreasing over $\reals^+$.
\end{lemma}

\begin{proof}
\emph{W.l.o.g} we can rename and order the pairs in $A$ so that:
\begin{displaymath}
\frac{v_1}{c_1}\geq \frac{v_2}{c_2}\geq\ldots\geq\frac{v_m}{c_m}.
\end{displaymath}
Then, $\mathcal{O}(A,b)$ has the following simple piecewise linear expression:
\begin{equation}\label{eq:pw}
    \mathcal{O}(A,b) = 
    \begin{cases}
        b\frac{v_1}{c_1}&\text{if }0\leq b<c_1\\
        \sum_{k=1}^{i-1}v_k + \left(b-\sum_{k=1}^{i-1}c_k\right)\frac{v_i}{c_i}
        &\text{if } \sum_{k=1}^{i-1}c_k\leq b< \sum_{k=1}^ic_k\\
        \sum_{k=1}^m v_k&\text{if } b\geq\sum_{i=1}^m c_k

    \end{cases}
\end{equation}

Let us define for $t\in\reals^+$, $n(t)\defeq \inf\Big\{i\text{ s.t.
} \sum_{k=1}^i c_k > t\Big \}$ with $n(t)=+\infty$ when the set is empty. In
particular, note that $x\mapsto n(t)$ is non-decreasing. Denoting
$\partial_+\mathcal{O}_A$ the right derivative of $\mathcal{O}(A,\cdot)$, one
can write $\partial_+\mathcal{O}_A(t)
=\frac{v_{n(t)}}{c_{n(t)}}$, with the convention that
$\frac{v_\infty}{c_\infty} = 0$.

Writing $i \defeq \sup\Big\{j\text{ s.t.
} \frac{v_j}{c_j}\geq\frac{v}{c}\Big\}$, it is easy to see that
$\partial_+\mathcal{O}_{A\cup\{x\}}\geq\partial_+\mathcal{O}_A$. Indeed:
\begin{enumerate}
    \item if $n(t)\leq i$ then $\partial_+\mathcal{O}_{A\cup\{x\}}(t)
        = \partial_+\mathcal{O}_A(t)= \frac{v_{n(t)}}{c_{n(t)}}$.
    \item if $n(t)\geq i+1$ and $n(t-c)\leq i$ then $\partial_+\mathcal{O}_{A\cup\{x\}}(t)
        = \frac{v}{c}\geq\frac{v_{n(t)}}{c_{n(t)}}= \partial_+\mathcal{O}_A(t)$.
    \item if $n(t-c)\geq i+1$, then $\partial_+\mathcal{O}_{A\cup\{x\}}
    = \frac{v_{n(t-c)}}{c_{n(t-c)}}\geq
    \frac{v_{n(t)}}{c_{n(t)}}=\partial_+\mathcal{O}_A(t)$.
\end{enumerate}

Let us now consider $b$ and $c$ such that $b\leq c$. Then, using the integral
representation of $\mathcal{O}(A\cup\{x\},\cdot)$ and $\mathcal{O}(A,\cdot)$, we get:
\begin{multline*}
    \mathcal{O}(A\cup\{x\},c)-\mathcal{O}(A\cup\{x\},b)=\int_b^c\partial_+\mathcal{O}_{A\cup\{x\}}(t)dt\\
    \geq\int_b^c\partial_+\mathcal{O}_A(t)dt=\mathcal{O}(A,c)-\mathcal{O}(A,b)
\end{multline*}
Re-ordering the terms, $\mathcal{O}(A\cup\{x\},c)-\mathcal{O}(A,c)\geq
\mathcal{O}(A\cup\{x\},b)-\mathcal{O}(A,b)$
which concludes the proof of the lemma.
\end{proof}

\begin{lemma}\label{lemma:sub}
    Let $b\in\reals^+$, then the function $A\mapsto \mathcal{O}(A,b)$ is submodular.
\end{lemma}

\begin{proof}
    Let $A\subseteq\mathcal{U}$ and $x, y\in\mathcal{U}\setminus A$. Using the
    second-order characterization of submodular functions, it suffices to show
    that:
    \begin{equation}\label{eq:so}
        \mathcal{O}(A\cup\{x\},b)-\mathcal{O}(A,b)\geq
        \mathcal{O}(A\cup\{y\}\cup\{x\},b)-\mathcal{O}(A\cup\{y\},b)
    \end{equation}

    Let us write $x=(v_x, c_x)$ and $y=(v_y,c_y)$. We distinguish two cases
    based on the relative position of $\frac{v_x}{c_x}$ and $\frac{v_y}{c_y}$.
    The following notations will be useful: $S_A^x \defeq \big\{(v,c)\in
    A\text{ s.t.  }\frac{v_x}{c_x}\leq\frac{v}{c}\big\}$ and $P_A^x\defeq
    A\setminus S_A^x$.

    \textbf{Case 1:} If $\frac{v_y}{c_y}\geq \frac{v_x}{c_x}$, then one can
    write:
    \begin{gather*}
        \mathcal{O}(A\cup\{y\}\cup\{x\},b) = \mathcal{O}(P_A^y\cup\{y\},b_1)+
        \mathcal{O}(S_A^y\cup\{x\},b_2)\\
        \mathcal{O}(A\cup\{y\},b) = \mathcal{O}(P_A^y\cup\{y\},b_1)
        + \mathcal{O}(S_A^y,b_2)
    \end{gather*}
    where $b_1$ is the fraction of the budget $b$ spent on $P_A^y\cup\{y\}$ and
    $b_2=b-b_1$.
    
    Similarly:
    \begin{gather*}
        \mathcal{O}(A\cup\{x\},b) = \mathcal{O}(P_A^y, c_1) + \mathcal{O}(S_A^y\cup\{x\},c_2)\\
        \mathcal{O}(A, b) = \mathcal{O}(P_A^y, c_1) + \mathcal{O}(S_A^y,c_2)
    \end{gather*}
    where $c_1$ is the fraction of the budget $b$ spent on $P_A^y$ and $c_2
    = b - c_1$. 

    Note that $b_1\geq c_1$: an optimal solution will first spent as much
    budget as possible on $P_A^y\cup\{y\}$ before adding elements in
    $S_A^y\cup\{x\}$. We can now rewrite \eqref{eq:so}:
    \begin{displaymath}
        \mathcal{O}(S_A^y\cup\{x\},c_2)+\mathcal{O}(S_A^y,c_2)\geq
        \mathcal{O}(S_A^y\cup\{x\},b_2)+\mathcal{O}(S_A^y,b_2)
    \end{displaymath}
    with $c_2\geq b_2$. This inequality is true by Lemma~\ref{lemma:nd}.

    \textbf{Case 2:} If $\frac{v_x}{c_x}\geq \frac{v_y}{c_y}$, we now decompose
    the solution on $P_A^x$ and $S_A^x$:
    \begin{gather*}
        \mathcal{O}(A\cup\{x\},b) = \mathcal{O}(P_A^x\cup\{x\},b_1)
        + \mathcal{O}(S_A^x,b_2)\\
        \mathcal{O}(A,b) = \mathcal{O}(P_A^x,c_1)+\mathcal{O}(S_A^x,c_2)
        \intertext{and}
        \mathcal{O}(A\cup\{y\}\cup\{x\},b) = \mathcal{O}(P_A^x\cup\{x\},b_1)
        + \mathcal{O}(S_A^x\cup\{y\},b_2)\\
        \mathcal{O}(A\cup\{y\},b) = \mathcal{O}(P_A^x,c_1)+\mathcal{O}(S_A^x\cup\{y\},c_2)
    \end{gather*}
    with $b_1+b_2=b$, $c_1+c_2=b$ and $b_2\leq c_2$. The equation \eqref{eq:so}
    can be rewritten:
    \begin{displaymath}
        \mathcal{O}(S_A^x,b_2)-\mathcal{O}(S_A^x,c_2)\geq
        \mathcal{O}(S_A^x\cup\{u\},b_2)-\mathcal{O}(S_A^x\cup\{y\},c_2)
    \end{displaymath}
    Reordering the terms, we get:
    \begin{displaymath}
        \mathcal{O}(S_A^x\cup\{y\},c_2) -\mathcal{O}(S_A^x,c_2)\geq
        \mathcal{O}(S_A^x\cup\{u\},b_2)-\mathcal{O}(S_A^x,b_2)
    \end{displaymath}
    with $c_2\geq b_2$. This is again true by Lemma~\ref{lemma:nd}.
\end{proof}

\begin{proposition}\label{prop:sub}
    Let $b\in\mathbf{R}^+$, then the function
    $S\mapsto\mathcal{O}(\neigh{S},b)$ is submodular over $2^X$.
\end{proposition}

\begin{proof}
    Let us consider $S$ and $T$ such that $S\subseteq T\subseteq X$ and $x\in
    X\setminus T$. In particular, note that $\neigh{S}\subseteq\neigh{T}$. 

    Let us write $\neigh{S\cup\{x\}}=\neigh{S}\cup R$ with $\neigh{S}\cap
    R=\emptyset$ and similarly, $\neigh{T\cup\{x\}}=\neigh{T}\cup R'$ with
    $\neigh{T}\cap R'=\emptyset$. Using these notations, it is clear that
    $R'\subseteq R$. Let us enumerate the elements in $R'$ using an arbitrary
    order: $R'=\{u_1,\ldots,u_k\}$. Then one can write:
    \begin{displaymath}
        \begin{split}
            \mathcal{O}(\neigh{T\cup\{x\}},b)&=\mathcal{O}(\neigh{T}\cup R',b)\\
            &=\sum_{i=1}^k\mathcal{O}(\neigh{T}\cup\{u_1,\ldots u_i\},b)
            -\mathcal{O}(\neigh{T}\cup\{u_1,\ldots u_{i-1}\},b)\\
            &\leq \sum_{i=1}^k\mathcal{O}(\neigh{S}\cup\{u_1,\ldots u_i\},b)
            -\mathcal{O}(\neigh{S}\cup\{u_1,\ldots u_{i-1}\},b)\\
            &=\mathcal{O}(\neigh{S}\cup R',b)-\mathcal{O}(\neigh{S},b)
        \end{split}
    \end{displaymath}
    where the inequality comes from the submodularity of
    $A\mapsto\mathcal{O}(A,b)$ proved in Lemma~\ref{lemma:sub}. This same
    function is also obviously set-increasing, hence:
    \begin{displaymath}
        \begin{split}
            \mathcal{O}(\neigh{S}\cup R',b)-\mathcal{O}(\neigh{S},b)
            &\leq \mathcal{O}(\neigh{S}\cup R,b)-\mathcal{O}(\neigh{S},b)\\
            &=\mathcal{O}(\neigh{S\cup\{x\}},b)-\mathcal{O}(\neigh{S},b)
        \end{split}
    \end{displaymath}
    This concludes the proof of the proposition.
\end{proof}

Using Proposition~\ref{prop:sub}, we can finally reduce \eqref{eq:sub} to
a submodular optimization problem by treating $|S|$ as a variable to optimize.
Specifically, \eqref{eq:sub} is equivalent to the following problem:
\begin{equation}\label{eq:sub2}
\begin{split}
    \max_{1\leq t\leq k}\max_{|S|\leq k-t}\mathcal{O}(\neigh{S},t)
\end{split}
\end{equation}

\begin{algorithm}
    \caption{Combinatorial algorithm}
    \label{alg:comb}
    \algsetup{indent=2em}
    \begin{algorithmic}[1]
        \STATE $S\leftarrow \emptyset$
        \FOR{$t=1$ \TO $k-1$}
            \STATE $S_t\leftarrow \emptyset$
            \FOR{$i=1$ \TO $k-t$}
                \STATE $x^*\leftarrow\argmax_{x\in
                X\setminus S_t}\mathcal{O}(\neigh{S_t\cup\{x\}},t)
                -\mathcal{O}(\neigh{S_t},t)$\label{line:argmax}
                \STATE $S_t\leftarrow S_t\cup\{x^*\}$
            \ENDFOR
            \IF{$\mathcal{O}(\neigh{S_t},t)>\mathcal{O}(\neigh{S},k-|S|)$}
                \STATE $S\leftarrow S_t$
            \ENDIF
        \ENDFOR
        \RETURN $S$
    \end{algorithmic}
\end{algorithm}

\begin{proposition}
    Let $S$ be the set computed by Algorithm~\ref{alg:comb} and let us denote
    by $A(S)$ the value of the adaptive policy which selects $S$ on the first
    day. Then:
    \begin{displaymath}
        A(S) \geq \frac{1}{2}\left(1-\frac{1}{e}\right)OPT.
    \end{displaymath}
    Furthermore, Algorithm~\ref{alg:comb} runs in time $O\left(n\log
    n+k^2m\min(\frac{k}{p_\text{min}},n)\right)$, where $m=|X|$, $n=|\neigh{X}|$
    and $p_\text{min} =\min\{p_u,u\in\neigh{X}\}$.
\end{proposition}

\begin{proof}
    For each realization of the neighbors $R$, let us define by $X_R$ the
    random set which includes each $u\in R$ with probability $q_u$ on the
    second day and:
    \begin{displaymath}
        \tilde{X}_R=
    \begin{cases}
        X_R&\text{if } |X_R|\leq t\\
        \emptyset&\text{otherwise}
    \end{cases}
    \end{displaymath}
    where $t = k-|S|$. $\tilde{X}_R$ is a random variable over the feasible
    solutions on the second day. As a consequence:
    \begin{displaymath}
        A(S) \geq \sum_{R\subseteq\neigh{S}} p_R\mathbb{E}\big[f(\tilde{X}_R)\big]
    \end{displaymath}
    Let us define by $Y$ the random set which includes each $u\in\neigh{X}$
    with probability $p_uq_u$ and:
    \begin{displaymath}
        \tilde{Y}=
        \begin{cases}
            Y&\text{if } |Y|\leq t\\
            \emptyset&\text{otherwise}
        \end{cases}.
    \end{displaymath}
    It is easy to see that $\tilde{X}_R$ is the conditional expectation of
    $\tilde{Y}$ given
    $R$. Hence:
    \begin{displaymath}
        \sum_{R\subseteq\neigh{S}} p_R\mathbb{E}\big[f(\tilde{X}_R)\big]
        = \mathbb{E}\big[f(\tilde{Y})\big]
    \end{displaymath}
    But:
    \begin{align*}
        \mathbb{E}\big[f(\tilde{Y})\big]
        &=\sum_{T\subseteq\neigh{X}}\mathbb{P}\big[\tilde{Y}=T\big]f(T)
    = \sum_{\substack{T\subseteq\neigh{X}\\|T|\leq
    t}}\mathbb{P}\big[Y=T\big]f(T)\\
    &= \mathbb{E}\big[f(Y)\big]-\sum_{\substack{T\subseteq\neigh{X}\\|T|>
    t}}\mathbb{P}\big[Y=T\big]f(T)
    \end{align*}
    Roughly:
    \begin{displaymath}
        \sum_{\substack{T\subseteq\neigh{X}\\|T|>
        t}}\mathbb{P}\big[Y=T\big]f(T)\leq
        \frac{1}{2}\mathbb{E}\big[f(Y)\big]
    \end{displaymath}
    Combining the above inequalities we get:
    \begin{displaymath}
        A(S)\geq \frac{1}{2} \mathbb{E}\big[f(Y)\big]
    \end{displaymath}
    But $\mathbb{E}\big[f(Y)\big]$ is precisely the value of the non-adaptive
    solution computed by Algorithm~\ref{alg:comb} which is
    a $\left(1-\frac{1}{e}\right)$ approximation of the
    optimal non adaptive solution. Hence:
    \begin{displaymath}
        A(S) \geq \frac{1}{2}\left(1-\frac{1}{e}\right) OPT_{NA}
    \end{displaymath}
    Finally, we conclude with Proposition~\ref{prop:gap}
\end{proof}

\subsection{From expectation to high probability}

\subsection{Questions}

\begin{itemize}
    \item Can it be parallelized (map-reduce)?
\end{itemize}